Numerical evaluation of spin foam amplitudes beyond simplices

Abstract

We present the first numerical calculation of the 4D Euclidean spin foam vertex amplitude for vertices with hypercubic combinatorics. Concretely, we compute the amplitude for coherent boundary data peaked on cuboid and frustum shapes. We present the numerical algorithms to explicitly compute the vertex amplitude and compare the results in different cases to the semi-classical approximation of the amplitude. Overall we find good qualitative agreement of the amplitudes and evidence of convergence of the asymptotic formula to the full amplitude already at fairly small spins, yet also differences in the frequency of oscillations and a phase shift absent in the 4-simplex case. However, due to rapidly growing numerical costs, we cannot reach sufficiently high spins to prove agreement of both amplitudes. Lastly, this setup allows us to explore non-uniform vertex amplitudes, where some representations are small while others are large; we find indications that scenarios might exist in which the semi-classical amplitude is a valid approximation even if some spins remain small. This suggests that the transition of the quantum to the semi-classical regime (for a single vertex amplitude) is intricate.

How-To compute EPRL spin foam amplitudes

“…where in the first equality we used (72) (in addition to the SL(2, C) irrep properties) and in the second one (69). Since there are no phases depending on the summed index, we conclude that the orientation of the (4jm) spins in the booster function (17) is irrelevant. This justifies the fact that we draw the latter without arrows.…”

“…The booster functions encode how the EPRL model imposes the quantum simplicity constraints and depend on the Immirzi parameter γ. Note that, in the definition (17), we dropped the information on the orientation of the faces. The orientation of the faces in the booster function is irrelevant.…”

“…sl2cfoam computes the booster functions performing a numerical integration of the boost matrix elements (17). The integrand is rewritten as a finite sum of exponentials with complex coefficients to tame its highly oscillating behaviour.…”

“…It was possible because of a paradigm shift in the field. It evolved from a theoretical framework to circumvent the difficulties in imposing the Hamiltonian constraint in the canonical approach [10] into a concrete tool where numerical calculations of transition amplitudes are possible [11,12,13,14,15,16,17].…”

“…where 2 F 1 is the Gauss hypergeometric function. The phase used in (70) is the same introduced in [31], which ensures the reality of the booster function (17). The reduced matrix elements (70) satisfy the following relation:…”